Maximizing weighted sums of binomial coefficients using generalized continued fractions

Abstract

Let $m,r\in\mathbb{Z}$ and $ω\in\mathbb{R}$ satisfy $0\leqslant r\leqslant m$ and $ω\geqslant1$. Our main result is a generalized continued fraction for an expression involving the partial binomial sum $s_m(r) = \sum_{i=0}^r\binom{m}{i}$. We apply this to create new upper and lower bounds for $s_m(r)$ and thus for $g_{ω,m}(r)=ω^{-r}s_m(r)$. We also bound an integer $r_0 \in \{0,1,\dots,m\}$ such that $g_{ω,m}(0)<\cdots \cdots>g_{ω,m}(m)$. For real $ω\geqslant\sqrt3$ we prove that $r_0\in\{\lfloor\frac{m+2}{ω+1}\rfloor,\lfloor\frac{m+2}{ω+1}\rfloor+1\}$, and also $r_0 =\lfloor\frac{m+2}{ω+1}\rfloor$ for $ω\in\{3,4,\dots\}$ or $ω=2$ and $3\nmid m$.